Unbinned Likelihood Analysis for X-ray Polarization

Abstract: We present a systematic study of the unbinned, photon-by-photon likelihood technique which can be used as an alternative method to analyse phase-dependent, X-ray spectro-polarimetric observations obtained with IXPE and other photo-electric polarimeters. We apply the unbinned technique to models of the luminous X-ray pulsar Hercules X-1, for which we produce simulated observations using ixpeobssim package. We consider minimal knowledge about the actual physical process responsible for the polarized emission from the accreting pulsar and assume that the observed phase-dependent polarization angle can be described by the rotating vector model. Using the unbinned technique, the detector's modulation factor, and the polarization information alone, we found that both the rotating vector model and the underlying spectro-polarimetry model can reconstruct equally well the geometric configuration angles of the accreting pulsar. However, the measured polarization fraction becomes biased with respect to underlying model unless the energy dispersion and effective area of the detector are also taken into account. To this end, we present an energy-dispersed likelihood estimator that is proved to be unbiased. For different analyses, we obtain posterior distributions from multiple ixpeobssim realizations and show that the unbinned technique yields $\sim 10\%$ smaller error bars than the binned technique. We also discuss alternative sources, such as magnetars, in which the unbinned technique and the rotating vector model might be applied.